The Khovanov homology of infinite braids

Abstract

We show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones–Wenzl projector. This result extends Lev Rozansky’s categorification of the Jones–Wenzl projectors using the limiting complex of infinite torus braids. We also show a similar result for the limiting Lipshitz–Sarkar–Khovanov stable homotopy types of the closures of such braids. Extensions to more general infinite braids are also considered.